How to checker deselect edge and vertex rings
Since my answer this similar question is somewhat outdated, here's a new answer.
Edge loops

Ensure you're in edge select mode (⎈ Ctrl↹ Tab)

Select an edge ring by pressing ⎈ Ctrl⎇ Alt RMB over an edge

Run checker deselect (the Space search is useful here)

Run multi select loops



Note that this selects edge loops as shown in your example. To do the same for edge rings, do the inverse:
Edge rings

Ensure you're still in edge select mode
Select an edge loop this time (⎇ Alt RMB)
Run checker deselect
Run multi select loops, then enable rings in the redo menu (F6 or 3D view &gt; Toolshelf)